I had a textbook hot-start IP issue and it never had anything to do with the engine running. I really doubt a running IP getting proper fuel flow would die on itself.
First I would check fuel pressure while running, the IP *will* pull it's own fuel but that's not a good way to make it around town.
Also, make sure your idle RPM is high enough. These are mechanically injected engines, its possible for the idle to be so low the engine won't run when hot, though I haven't heard of it yet.
As
@david85 said a sticking metering valve is a semi-common issue with pumps that have sat for a long time. The ATF/Diesel Kleen treatment in the fuel filter works very well in this case.